Electromagnetic Waves — Given E, find B....

[astrolady022]

Homework Statement

A plane electromagnetic wave travels upward. At t = 0, x = 0, its electric field has the value E = 5 V/m and points eastward. What is the wave's magnetic field at t = 0, x = 0?

Homework Equations

B=B init. sin(kx-wt)
E=E inti. sin(kx-wt)
E=cB

The Attempt at a Solution

I am pretty stumped on this one. I tried substituting E=5v into the third equation and solving for B, but that doesn't seem like the correct approach. Equations one and two don't seem to be much help since I am only told about what is happening when x and t both =0. Any help here is appreciated.

 

[rude man]

How do E and B relate as a function of the speed of light?

 

[Chandra Prayaga]

The value of E at x = 0, and t = 0 is 5 V/m. That means the electric field cannot be described by the sine function that you wrote.

 

[astrolady022]

The value of E at x = 0, and t = 0 is 5 V/m. That means the electric field cannot be described by the sine function that you wrote.

Is there another equation I should be using? I cannot think of any other EM wave equations I have learned other than Maxwell's. And I'm not sure how to manipulate the equations to work with the problem since I don't have a lot of information.

 

[astrolady022]

How do E and B relate as a function of the speed of light?

Binit /E init =c. This is the other equation I have been trying to manipulate.

 

[Chandra Prayaga]

Is there another equation I should be using? I cannot think of any other EM wave equations I have learned other than Maxwell's. And I'm not sure how to manipulate the equations to work with the problem since I don't have a lot of information.

A cosine function is a possibility.

 

[astrolady022]

A cosine function is a possibility.

ok. I could take the derivative of the B function with respect to time to get
flux=-Bint w cos(kx-wt)
Substituting my values for x and t I get
flux=-wB
From here, I could use that the flux = the integral of B dot da. Here, I can't find a way to relate the equation back to the electric field.
My other option is to look at the derivative of the E function. Integrating with respect to time, I get
dE/dT=-Eint w cos(kx-wt)
I can use this equation to solve the problem if I assume dE/dT=B. However, I am not sure I can make this assumption. If I do, I can simply substitute in the values from the problem and get an answer. Let me know if either of these approaches make sense to you.

 

[Ray Vickson]

Homework Statement

A plane electromagnetic wave travels upward. At t = 0, x = 0, its electric field has the value E = 5 V/m and points eastward. What is the wave's magnetic field at t = 0, x = 0?

Homework Equations

B=B init. sin(kx-wt)
E=E inti. sin(kx-wt)
E=cB

The Attempt at a Solution

I am pretty stumped on this one. I tried substituting E=5v into the third equation and solving for B, but that doesn't seem like the correct approach. Equations one and two don't seem to be much help since I am only told about what is happening when x and t both =0. Any help here is appreciated.

Who says that $B = B_{\text{init}} \sin(kx - wt)$, etc? Isn't a more solution to Maxwell's equations given by
$$ \vec{E} = \vec{E}_1 \cos(kx - wt) + \vec{E}_2 \sin(kx - wt) \\
\vec{B} = \vec{B}_1 \cos(kx - wt) + \vec{B}_2 \sin(kx - wt),$$ etc?

 

[astrolady022]

Who says that $B = B_{\text{init}} \sin(kx - wt)$, etc? Isn't a more solution to Maxwell's equations given by
$$ \vec{E} = \vec{E}_1 \cos(kx - wt) + \vec{E}_2 \sin(kx - wt) \\
\vec{B} = \vec{B}_1 \cos(kx - wt) + \vec{B}_2 \sin(kx - wt),$$ etc?

What I was using was a wave equation I found in a textbook. I'm not sure how your equations would apply to this situation since the problem only looks at a certain instance in time and not a change. Maybe I should be looking outside of Maxwell's equations?

 

[Chandra Prayaga]

I believe the point that Ray Vickson is making above is that the sine function that you used originally for the electric (and magnetic) field is not general enough. The wave equation that you found in a textbook certainly does not cover the situation where E ≠ 0 for x = 0 and t = 0. That is why I suggested the cosine function for the fields (not their derivatives) as a possibility.
Also, in post #7, you wrote:
flux = dB/dt. Where did you see that equation, and it is the flux of what?
It is not that you should look "outside of" Maxwell equations. Every EM wave has to obey Maxwell equations. You should be careful about particular cases when you look at some equation in some textbook.

 

[astrolady022]

I thought that a changing magnetic field was a flux, so dB/dt= flux. But I will make note that that is incorrect. As far as the cosine equation, it is not one I have seen before. Looking at the equations Ray Vickson has posted, if I substituted in what I know, which is that, at this instant, both x and t are 0, it seems like both of the sines will become 0 and both of the cosines will become one. Then, I am just left with E=E and B=B. So I'm not sure what needs to happen from there. Is there another way that I need to manipulate these equations to make them make sense? Also thanks to all of you for working with me here. I really appreciate it.

 

[vela]

You're making the problem too complicated because you're trying to use formulas. The question is really more of a conceptual one. For an electromagnetic wave, how are the electric and magnetic field related? What's their relative phase? What about the directions of the fields and the direction the wave is moving?

 

[Chandra Prayaga]

It looks like you are not clear about the meaning of the equations that you are using. When you gave relevant equations
B=B init. sin(kx-wt)
E=E inti. sin(kx-wt)
E=cB
How did you get these equations? What does the init mean in the equations?

 

[astrolady022]

You're making the problem too complicated because you're trying to use formulas. The question is really more of a conceptual one. For an electromagnetic wave, how are the electric and magnetic field related? What's their relative phase? What about the directions of the fields and the direction the wave is moving?

Ok I see. We have not yet covered relative phases. It is coming up in a few lectures. I know the electric and magnetic portions of the wave oscillate, but at this instant, the electric portion points East. I also know the magnetic portion must lie at a 90 degree angle to the electric field. So this gives me a picture of what the magnetic field will look like but the question is asking for a value I am not sure how to find.

 

[astrolady022]

It looks like you are not clear about the meaning of the equations that you are using. When you gave relevant equations
B=B init. sin(kx-wt)
E=E inti. sin(kx-wt)
E=cB
How did you get these equations? What does the init mean in the equations?

Here, the init. stands for initial. I am looking through my textbook for the origin of the equations. I think it was from a different form of the equations or possibly a derivation I am taking out of context.

 

[Ray Vickson]

What I was using was a wave equation I found in a textbook. I'm not sure how your equations would apply to this situation since the problem only looks at a certain instance in time and not a change. Maybe I should be looking outside of Maxwell's equations?

Well, one simple solution would be $E = E_0 \cos(kx-wt),\: B = B_0 \cos(kx - wt),$ while another simple solution would be $E = E_1 \sin(kx-wt),\: B = B_1 \sin(kx -wt)$. A more general solution would just be the sum of those two simple solutions.

 

[Chandra Prayaga]

Look for an oscillating function which does not become zero at x = 0 and t = 0, for example, the cosine function, as suggested in post #16.
There are many details about the EM waves that we must be careful about. For example, you stated above that the magnetic field is at 90 degrees to the electric field. That is not enough to determine the direction of B. If E is pointing East, according to your statement, B could be pointing north, south, up, or down. Which is it, and why?

 

[astrolady022]

Well, one simple solution would be $E = E_0 \cos(kx-wt),\: B = B_0 \cos(kx - wt),$ while another simple solution would be $E = E_1 \sin(kx-wt),\: B = B_1 \sin(kx -wt)$. A more general solution would just be the sum of those two simple solutions.

Oh ok. It looks like that simple solution was what I was trying to use and causing confusion with. Thanks for clarifying that for me. So let's say I use an approach using these equations rather than a more conceptual one. Should I work with these more simplified equations to find a solution to the problem? Does that make more sense?

 

[astrolady022]

Look for an oscillating function which does not become zero at x = 0 and t = 0, for example, the cosine function, as suggested in post #16.
There are many details about the EM waves that we must be careful about. For example, you stated above that the magnetic field is at 90 degrees to the electric field. That is not enough to determine the direction of B. If E is pointing East, according to your statement, B could be pointing north, south, up, or down. Which is it, and why?

That part is confusing me. I am not told what direction the wave is moving with respect to the electric field. I am simply told it is moving upward. I cannot assume that this is in a Northern or Eastern direction. So without knowingthe direction of the wave I am not sure how I can determine the direction of the magnetic field in relation to the wave.

 

[rude man]

That part is confusing me. I am not told what direction the wave is moving with respect to the electric field. I am simply told it is moving upward. I cannot assume that this is in a Northern or Eastern direction. So without knowingthe direction of the wave I am not sure how I can determine the direction of the magnetic field in relation to the wave.

The direction of the B vector is perpendicular to both the direction of propagation and the direction of the E vector. The relation is P = E x B. P is power in magnitude and direction of propagation. Don't worry about the meaning of power right now, just use the formula to give you the relative orientations of the E, B and P directions. This is the "right-hand rule".

 

[ehild]

That part is confusing me. I am not told what direction the wave is moving with respect to the electric field. I am simply told it is moving upward. I cannot assume that this is in a Northern or Eastern direction. So without knowingthe direction of the wave I am not sure how I can determine the direction of the magnetic field in relation to the wave.

"Upward" is the direction of propagation. You can choose one axis of your coordinate system pointing upward. The other two axes can point to East and to North. I think you have learned some right - hand rule for the directions of the electric and magnetic fields and the direction of propagation for a wave?

 

[vela]

I know the electric and magnetic portions of the wave oscillate, but at this instant, the electric portion points East. I also know the magnetic portion must lie at a 90 degree angle to the electric field. So this gives me a picture of what the magnetic field will look like but the question is asking for a value I am not sure how to find.

I hope you've seen a figure like the one below and discussed in class the basic facts about electromagnetic plane waves depicted in the diagram.

From the diagram, you can see the relationship between the directions of $\vec{E}$ and $\vec{B}$ and the direction the wave is moving. Use the information given in the problem to determine the direction of $\vec{B}$.

From the diagram, you can see that the electric and magnetic fields are in phase. At any given time, the fields are zero at the same points in space; they're maximums at the same points in space; they're half their maximum values at the same points in space; and so on. In other words, at a given place and time, the magnitude of the magnetic field is proportional to the magnitude of the electric field. What is the constant of proportionality? (Hint: look at your relevant equations.) So what's the magnitude of $\vec{B}(x=0,t=0)$?